Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.4%

Time: 12.4s

Alternatives: 13

Speedup: 1.4×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* 2.0 (* J (log1p (expm1 (* l (cos (* K 0.5))))))) U))

C

double code(double J, double l, double K, double U) {
	return (2.0 * (J * log1p(expm1((l * cos((K * 0.5))))))) + U;
}

Java

public static double code(double J, double l, double K, double U) {
	return (2.0 * (J * Math.log1p(Math.expm1((l * Math.cos((K * 0.5))))))) + U;
}

Python

def code(J, l, K, U):
	return (2.0 * (J * math.log1p(math.expm1((l * math.cos((K * 0.5))))))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(2.0 * Float64(J * log1p(expm1(Float64(l * cos(Float64(K * 0.5))))))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * N[(J * N[Log[1 + N[(Exp[N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 86.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 63.3%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. log1p-expm1-u 99.8%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

   2. *-commutative 99.8%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right)\right)\right) + U \]

5. Applied egg-rr 99.8%

   \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) + U \]

6. Final simplification 99.8%

   \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \]
7. Add Preprocessing

Alternative 2: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (* (* J (* (pow l 7.0) 0.0003968253968253968)) (cos (/ K 2.0))))))
   (if (<= l -4.0)
     t_0
     (if (<= l 3400.0)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 6.8e+43) (+ U (* J (- (exp l) (exp (- l))))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (pow(l, 7.0) * 0.0003968253968253968)) * cos((K / 2.0)));
	double tmp;
	if (l <= -4.0) {
		tmp = t_0;
	} else if (l <= 3400.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 6.8e+43) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((j * ((l ** 7.0d0) * 0.0003968253968253968d0)) * cos((k / 2.0d0)))
    if (l <= (-4.0d0)) then
        tmp = t_0
    else if (l <= 3400.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 6.8d+43) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (Math.pow(l, 7.0) * 0.0003968253968253968)) * Math.cos((K / 2.0)));
	double tmp;
	if (l <= -4.0) {
		tmp = t_0;
	} else if (l <= 3400.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 6.8e+43) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * (math.pow(l, 7.0) * 0.0003968253968253968)) * math.cos((K / 2.0)))
	tmp = 0
	if l <= -4.0:
		tmp = t_0
	elif l <= 3400.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 6.8e+43:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * Float64((l ^ 7.0) * 0.0003968253968253968)) * cos(Float64(K / 2.0))))
	tmp = 0.0
	if (l <= -4.0)
		tmp = t_0;
	elseif (l <= 3400.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 6.8e+43)
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * ((l ^ 7.0) * 0.0003968253968253968)) * cos((K / 2.0)));
	tmp = 0.0;
	if (l <= -4.0)
		tmp = t_0;
	elseif (l <= 3400.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 6.8e+43)
		tmp = U + (J * (exp(l) - exp(-l)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[(N[Power[l, 7.0], $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.0], t$95$0, If[LessEqual[l, 3400.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.8e+43], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4 or 6.80000000000000024e43 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 95.0%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{7}\right) \cdot 0.0003968253968253968\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 95.0%

         \[\leadsto \color{blue}{\left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 95.0%

      \[\leadsto \color{blue}{\left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -4 < l < 3400

   1. Initial program 73.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 98.8%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if 3400 < l < 6.80000000000000024e43

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 84.6%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;U + \left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+44}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (* (* J (* (pow l 7.0) 0.0003968253968253968)) (cos (/ K 2.0))))))
   (if (<= l -5.5)
     t_0
     (if (<= l 3400.0)
       (+
        U
        (*
         J
         (* l (* (cos (* K 0.5)) (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
       (if (<= l 1.2e+44) (+ U (* J (- (exp l) (exp (- l))))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (pow(l, 7.0) * 0.0003968253968253968)) * cos((K / 2.0)));
	double tmp;
	if (l <= -5.5) {
		tmp = t_0;
	} else if (l <= 3400.0) {
		tmp = U + (J * (l * (cos((K * 0.5)) * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else if (l <= 1.2e+44) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((j * ((l ** 7.0d0) * 0.0003968253968253968d0)) * cos((k / 2.0d0)))
    if (l <= (-5.5d0)) then
        tmp = t_0
    else if (l <= 3400.0d0) then
        tmp = u + (j * (l * (cos((k * 0.5d0)) * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    else if (l <= 1.2d+44) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (Math.pow(l, 7.0) * 0.0003968253968253968)) * Math.cos((K / 2.0)));
	double tmp;
	if (l <= -5.5) {
		tmp = t_0;
	} else if (l <= 3400.0) {
		tmp = U + (J * (l * (Math.cos((K * 0.5)) * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else if (l <= 1.2e+44) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * (math.pow(l, 7.0) * 0.0003968253968253968)) * math.cos((K / 2.0)))
	tmp = 0
	if l <= -5.5:
		tmp = t_0
	elif l <= 3400.0:
		tmp = U + (J * (l * (math.cos((K * 0.5)) * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	elif l <= 1.2e+44:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * Float64((l ^ 7.0) * 0.0003968253968253968)) * cos(Float64(K / 2.0))))
	tmp = 0.0
	if (l <= -5.5)
		tmp = t_0;
	elseif (l <= 3400.0)
		tmp = Float64(U + Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	elseif (l <= 1.2e+44)
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * ((l ^ 7.0) * 0.0003968253968253968)) * cos((K / 2.0)));
	tmp = 0.0;
	if (l <= -5.5)
		tmp = t_0;
	elseif (l <= 3400.0)
		tmp = U + (J * (l * (cos((K * 0.5)) * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	elseif (l <= 1.2e+44)
		tmp = U + (J * (exp(l) - exp(-l)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[(N[Power[l, 7.0], $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.5], t$95$0, If[LessEqual[l, 3400.0], N[(U + N[(J * N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+44], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5 or 1.20000000000000007e44 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 95.0%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{7}\right) \cdot 0.0003968253968253968\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 95.0%

         \[\leadsto \color{blue}{\left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 95.0%

      \[\leadsto \color{blue}{\left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -5.5 < l < 3400

   1. Initial program 73.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 99.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around 0 98.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.3333333333333333 \cdot {\ell}^{2}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot 0.3333333333333333}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 98.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot 0.3333333333333333}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. Taylor expanded in J around 0 99.0%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)} + U \]

   if 3400 < l < 1.20000000000000007e44

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 84.6%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.5:\\ \;\;\;\;U + \left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq 3400:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+44}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left({\ell}^{7} \cdot 0.0003968253968253968\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8500000000 \lor \neg \left(\ell \leq 3400\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -8500000000.0) (not (<= l 3400.0)))
   (+ U (* J (- (exp l) (exp (- l)))))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -8500000000.0) || !(l <= 3400.0)) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-8500000000.0d0)) .or. (.not. (l <= 3400.0d0))) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -8500000000.0) || !(l <= 3400.0)) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -8500000000.0) or not (l <= 3400.0):
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -8500000000.0) || !(l <= 3400.0))
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -8500000000.0) || ~((l <= 3400.0)))
		tmp = U + (J * (exp(l) - exp(-l)));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -8500000000.0], N[Not[LessEqual[l, 3400.0]], $MachinePrecision]], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8.5e9 or 3400 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 82.5%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]

   if -8.5e9 < l < 3400

   1. Initial program 74.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 97.3%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8500000000 \lor \neg \left(\ell \leq 3400\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \frac{2}{\frac{U}{\ell}}\\ \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;U \cdot \left(1 + \left|t\_0\right|\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (/ 2.0 (/ U l)))))
   (if (<= (cos (/ K 2.0)) -0.05) (* U (+ 1.0 (fabs t_0))) (* U (+ 1.0 t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 / (U / l));
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = U * (1.0 + fabs(t_0));
	} else {
		tmp = U * (1.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (2.0d0 / (u / l))
    if (cos((k / 2.0d0)) <= (-0.05d0)) then
        tmp = u * (1.0d0 + abs(t_0))
    else
        tmp = u * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 / (U / l));
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.05) {
		tmp = U * (1.0 + Math.abs(t_0));
	} else {
		tmp = U * (1.0 + t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 / (U / l))
	tmp = 0
	if math.cos((K / 2.0)) <= -0.05:
		tmp = U * (1.0 + math.fabs(t_0))
	else:
		tmp = U * (1.0 + t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(2.0 / Float64(U / l)))
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = Float64(U * Float64(1.0 + abs(t_0)));
	else
		tmp = Float64(U * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (2.0 / (U / l));
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.05)
		tmp = U * (1.0 + abs(t_0));
	else
		tmp = U * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(2.0 / N[(U / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(U * N[(1.0 + N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 84.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 70.5%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 70.5%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 70.5%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 70.5%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 78.5%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 78.5%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 78.5%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 40.2%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 40.2%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 40.2%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 21.0%

          \[\leadsto U \cdot \left(1 + \color{blue}{\sqrt{2 \cdot \left(J \cdot \frac{\ell}{U}\right)} \cdot \sqrt{2 \cdot \left(J \cdot \frac{\ell}{U}\right)}}\right) \]

       2. sqrt-unprod 57.8%

          \[\leadsto U \cdot \left(1 + \color{blue}{\sqrt{\left(2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right) \cdot \left(2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)}}\right) \]

       3. *-commutative 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{\color{blue}{\left(\left(J \cdot \frac{\ell}{U}\right) \cdot 2\right)} \cdot \left(2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)}\right) \]

       4. *-commutative 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{\left(\left(J \cdot \frac{\ell}{U}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(J \cdot \frac{\ell}{U}\right) \cdot 2\right)}}\right) \]

       5. swap-sqr 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{\color{blue}{\left(\left(J \cdot \frac{\ell}{U}\right) \cdot \left(J \cdot \frac{\ell}{U}\right)\right) \cdot \left(2 \cdot 2\right)}}\right) \]

       6. pow2 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{\color{blue}{{\left(J \cdot \frac{\ell}{U}\right)}^{2}} \cdot \left(2 \cdot 2\right)}\right) \]

       7. metadata-eval 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{{\left(J \cdot \frac{\ell}{U}\right)}^{2} \cdot \color{blue}{4}}\right) \]

   13. Applied egg-rr 57.8%

       \[\leadsto U \cdot \left(1 + \color{blue}{\sqrt{{\left(J \cdot \frac{\ell}{U}\right)}^{2} \cdot 4}}\right) \]
   14. Step-by-step derivation

       1. *-commutative 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{\color{blue}{4 \cdot {\left(J \cdot \frac{\ell}{U}\right)}^{2}}}\right) \]

       2. metadata-eval 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{\color{blue}{\left(2 \cdot 2\right)} \cdot {\left(J \cdot \frac{\ell}{U}\right)}^{2}}\right) \]

       3. unpow2 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{\left(2 \cdot 2\right) \cdot \color{blue}{\left(\left(J \cdot \frac{\ell}{U}\right) \cdot \left(J \cdot \frac{\ell}{U}\right)\right)}}\right) \]

       4. swap-sqr 57.8%

          \[\leadsto U \cdot \left(1 + \sqrt{\color{blue}{\left(2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right) \cdot \left(2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)}}\right) \]

       5. rem-sqrt-square 56.7%

          \[\leadsto U \cdot \left(1 + \color{blue}{\left|2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right|}\right) \]

       6. associate-*r* 56.7%

          \[\leadsto U \cdot \left(1 + \left|\color{blue}{\left(2 \cdot J\right) \cdot \frac{\ell}{U}}\right|\right) \]

       7. associate-*r/ 56.7%

          \[\leadsto U \cdot \left(1 + \left|\color{blue}{\frac{\left(2 \cdot J\right) \cdot \ell}{U}}\right|\right) \]

       8. associate-*l/ 56.3%

          \[\leadsto U \cdot \left(1 + \left|\color{blue}{\frac{2 \cdot J}{U} \cdot \ell}\right|\right) \]

       9. associate-/r/ 56.7%

          \[\leadsto U \cdot \left(1 + \left|\color{blue}{\frac{2 \cdot J}{\frac{U}{\ell}}}\right|\right) \]

       10. *-commutative 56.7%

           \[\leadsto U \cdot \left(1 + \left|\frac{\color{blue}{J \cdot 2}}{\frac{U}{\ell}}\right|\right) \]

       11. associate-/l* 56.7%

           \[\leadsto U \cdot \left(1 + \left|\color{blue}{J \cdot \frac{2}{\frac{U}{\ell}}}\right|\right) \]

   15. Simplified 56.7%

       \[\leadsto U \cdot \left(1 + \color{blue}{\left|J \cdot \frac{2}{\frac{U}{\ell}}\right|}\right) \]

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 61.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 61.2%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 61.2%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 61.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 63.8%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 68.7%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 68.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 61.5%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 66.3%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 66.3%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   12. Taylor expanded in J around 0 61.5%

       \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \frac{J \cdot \ell}{U}}\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 57.7%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(\frac{J}{U} \cdot \ell\right)}\right) \]

       2. associate-/r/ 65.8%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J}{\frac{U}{\ell}}}\right) \]

       3. associate-/l* 65.8%

          \[\leadsto U \cdot \left(1 + \color{blue}{\frac{2 \cdot J}{\frac{U}{\ell}}}\right) \]

       4. *-commutative 65.8%

          \[\leadsto U \cdot \left(1 + \frac{\color{blue}{J \cdot 2}}{\frac{U}{\ell}}\right) \]

       5. associate-/l* 66.3%

          \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{2}{\frac{U}{\ell}}}\right) \]

   14. Simplified 66.3%

       \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{2}{\frac{U}{\ell}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;U \cdot \left(1 + \left|J \cdot \frac{2}{\frac{U}{\ell}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{2}{\frac{U}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \cos \left(K \cdot 0.5\right)\\ t_1 := J \cdot \left(\frac{U}{J} + 2 \cdot t\_0\right)\\ \mathbf{if}\;\ell \leq -4000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 65000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+205}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (cos (* K 0.5)))) (t_1 (* J (+ (/ U J) (* 2.0 t_0)))))
   (if (<= l -4000000000000.0)
     t_1
     (if (<= l 65000000.0)
       (+ U (* 2.0 (* J t_0)))
       (if (<= l 6.2e+205) t_1 (* U (+ 1.0 (* 2.0 (* J (/ l U))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * cos((K * 0.5));
	double t_1 = J * ((U / J) + (2.0 * t_0));
	double tmp;
	if (l <= -4000000000000.0) {
		tmp = t_1;
	} else if (l <= 65000000.0) {
		tmp = U + (2.0 * (J * t_0));
	} else if (l <= 6.2e+205) {
		tmp = t_1;
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = l * cos((k * 0.5d0))
    t_1 = j * ((u / j) + (2.0d0 * t_0))
    if (l <= (-4000000000000.0d0)) then
        tmp = t_1
    else if (l <= 65000000.0d0) then
        tmp = u + (2.0d0 * (j * t_0))
    else if (l <= 6.2d+205) then
        tmp = t_1
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = l * Math.cos((K * 0.5));
	double t_1 = J * ((U / J) + (2.0 * t_0));
	double tmp;
	if (l <= -4000000000000.0) {
		tmp = t_1;
	} else if (l <= 65000000.0) {
		tmp = U + (2.0 * (J * t_0));
	} else if (l <= 6.2e+205) {
		tmp = t_1;
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * math.cos((K * 0.5))
	t_1 = J * ((U / J) + (2.0 * t_0))
	tmp = 0
	if l <= -4000000000000.0:
		tmp = t_1
	elif l <= 65000000.0:
		tmp = U + (2.0 * (J * t_0))
	elif l <= 6.2e+205:
		tmp = t_1
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * cos(Float64(K * 0.5)))
	t_1 = Float64(J * Float64(Float64(U / J) + Float64(2.0 * t_0)))
	tmp = 0.0
	if (l <= -4000000000000.0)
		tmp = t_1;
	elseif (l <= 65000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * t_0)));
	elseif (l <= 6.2e+205)
		tmp = t_1;
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * cos((K * 0.5));
	t_1 = J * ((U / J) + (2.0 * t_0));
	tmp = 0.0;
	if (l <= -4000000000000.0)
		tmp = t_1;
	elseif (l <= 65000000.0)
		tmp = U + (2.0 * (J * t_0));
	elseif (l <= 6.2e+205)
		tmp = t_1;
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[(N[(U / J), $MachinePrecision] + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4000000000000.0], t$95$1, If[LessEqual[l, 65000000.0], N[(U + N[(2.0 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e+205], t$95$1, N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4e12 or 6.5e7 < l < 6.20000000000000035e205

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 23.9%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 23.9%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 23.9%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 23.9%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in J around inf 38.9%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   if -4e12 < l < 6.5e7

   1. Initial program 74.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 95.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if 6.20000000000000035e205 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 50.4%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 50.4%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 50.4%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 50.4%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 62.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 87.8%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 87.8%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 54.0%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 79.1%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 79.1%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4000000000000:\\ \;\;\;\;J \cdot \left(\frac{U}{J} + 2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 65000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+205}:\\ \;\;\;\;J \cdot \left(\frac{U}{J} + 2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-239}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{elif}\;\frac{K}{2} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{2}{\frac{U}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-239)
   (* J (+ (* 2.0 l) (/ U J)))
   (if (<= (/ K 2.0) 2e-21)
     (* U (+ 1.0 (* J (/ 2.0 (/ U l)))))
     (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-239) {
		tmp = J * ((2.0 * l) + (U / J));
	} else if ((K / 2.0) <= 2e-21) {
		tmp = U * (1.0 + (J * (2.0 / (U / l))));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 5d-239) then
        tmp = j * ((2.0d0 * l) + (u / j))
    else if ((k / 2.0d0) <= 2d-21) then
        tmp = u * (1.0d0 + (j * (2.0d0 / (u / l))))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-239) {
		tmp = J * ((2.0 * l) + (U / J));
	} else if ((K / 2.0) <= 2e-21) {
		tmp = U * (1.0 + (J * (2.0 / (U / l))));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-239:
		tmp = J * ((2.0 * l) + (U / J))
	elif (K / 2.0) <= 2e-21:
		tmp = U * (1.0 + (J * (2.0 / (U / l))))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-239)
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	elseif (Float64(K / 2.0) <= 2e-21)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(2.0 / Float64(U / l)))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-239)
		tmp = J * ((2.0 * l) + (U / J));
	elseif ((K / 2.0) <= 2e-21)
		tmp = U * (1.0 + (J * (2.0 / (U / l))));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-239], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(K / 2.0), $MachinePrecision], 2e-21], N[(U * N[(1.0 + N[(J * N[(2.0 / N[(U / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 5e-239

   1. Initial program 83.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 64.8%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 64.8%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 64.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 67.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 71.4%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 71.4%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 55.5%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 59.3%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 59.3%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   12. Taylor expanded in J around inf 55.7%

       \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]

   if 5e-239 < (/.f64 K #s(literal 2 binary64)) < 1.99999999999999982e-21

   1. Initial program 89.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 61.9%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 61.9%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 61.9%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 61.9%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 66.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 73.8%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 73.8%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 66.6%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 73.8%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 73.8%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   12. Taylor expanded in J around 0 66.6%

       \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \frac{J \cdot \ell}{U}}\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 61.8%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(\frac{J}{U} \cdot \ell\right)}\right) \]

       2. associate-/r/ 73.8%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J}{\frac{U}{\ell}}}\right) \]

       3. associate-/l* 73.8%

          \[\leadsto U \cdot \left(1 + \color{blue}{\frac{2 \cdot J}{\frac{U}{\ell}}}\right) \]

       4. *-commutative 73.8%

          \[\leadsto U \cdot \left(1 + \frac{\color{blue}{J \cdot 2}}{\frac{U}{\ell}}\right) \]

       5. associate-/l* 73.8%

          \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{2}{\frac{U}{\ell}}}\right) \]

   14. Simplified 73.8%

       \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{2}{\frac{U}{\ell}}}\right) \]

   if 1.99999999999999982e-21 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 93.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 60.6%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-239}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{elif}\;\frac{K}{2} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{2}{\frac{U}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-239}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-239)
   (* J (+ (* 2.0 l) (/ U J)))
   (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-239) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 5d-239) then
        tmp = j * ((2.0d0 * l) + (u / j))
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-239) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-239:
		tmp = J * ((2.0 * l) + (U / J))
	else:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-239)
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-239)
		tmp = J * ((2.0 * l) + (U / J));
	else
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-239], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 5e-239

   1. Initial program 83.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 64.8%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 64.8%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 64.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 67.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 71.4%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 71.4%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 55.5%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 59.3%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 59.3%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   12. Taylor expanded in J around inf 55.7%

       \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]

   if 5e-239 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 92.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 61.1%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 61.1%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 61.1%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 66.5%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 70.1%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 70.1%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-239}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4300000000000 \lor \neg \left(\ell \leq 65000000\right):\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4300000000000.0) (not (<= l 65000000.0)))
   (* J (+ (* 2.0 l) (/ U J)))
   (+ U (* J (* 2.0 l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4300000000000.0) || !(l <= 65000000.0)) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U + (J * (2.0 * l));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4300000000000.0d0)) .or. (.not. (l <= 65000000.0d0))) then
        tmp = j * ((2.0d0 * l) + (u / j))
    else
        tmp = u + (j * (2.0d0 * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4300000000000.0) || !(l <= 65000000.0)) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U + (J * (2.0 * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4300000000000.0) or not (l <= 65000000.0):
		tmp = J * ((2.0 * l) + (U / J))
	else:
		tmp = U + (J * (2.0 * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4300000000000.0) || !(l <= 65000000.0))
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4300000000000.0) || ~((l <= 65000000.0)))
		tmp = J * ((2.0 * l) + (U / J));
	else
		tmp = U + (J * (2.0 * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4300000000000.0], N[Not[LessEqual[l, 65000000.0]], $MachinePrecision]], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.3e12 or 6.5e7 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 28.8%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 28.8%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 28.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 28.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 38.8%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 46.6%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 46.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 29.5%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 37.4%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 37.4%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   12. Taylor expanded in J around inf 35.6%

       \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]

   if -4.3e12 < l < 6.5e7

   1. Initial program 74.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 95.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 95.2%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 95.1%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in K around 0 83.5%

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. *-commutative 83.5%

         \[\leadsto U + \color{blue}{\left(J \cdot \ell\right) \cdot 2} \]

      2. associate-*r* 83.5%

         \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

   8. Simplified 83.5%

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4300000000000 \lor \neg \left(\ell \leq 65000000\right):\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-239}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{2}{\frac{U}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-239)
   (* J (+ (* 2.0 l) (/ U J)))
   (* U (+ 1.0 (* J (/ 2.0 (/ U l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-239) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U * (1.0 + (J * (2.0 / (U / l))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 5d-239) then
        tmp = j * ((2.0d0 * l) + (u / j))
    else
        tmp = u * (1.0d0 + (j * (2.0d0 / (u / l))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-239) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U * (1.0 + (J * (2.0 / (U / l))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-239:
		tmp = J * ((2.0 * l) + (U / J))
	else:
		tmp = U * (1.0 + (J * (2.0 / (U / l))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-239)
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(2.0 / Float64(U / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-239)
		tmp = J * ((2.0 * l) + (U / J));
	else
		tmp = U * (1.0 + (J * (2.0 / (U / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-239], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(J * N[(2.0 / N[(U / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 5e-239

   1. Initial program 83.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 64.8%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 64.8%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 64.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 67.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 71.4%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 71.4%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 55.5%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 59.3%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 59.3%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   12. Taylor expanded in J around inf 55.7%

       \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]

   if 5e-239 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 92.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 61.1%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 61.1%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 61.1%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 66.5%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 70.1%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 70.1%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 58.3%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 62.0%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 62.0%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   12. Taylor expanded in J around 0 58.3%

       \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \frac{J \cdot \ell}{U}}\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 56.2%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(\frac{J}{U} \cdot \ell\right)}\right) \]

       2. associate-/r/ 61.1%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J}{\frac{U}{\ell}}}\right) \]

       3. associate-/l* 61.1%

          \[\leadsto U \cdot \left(1 + \color{blue}{\frac{2 \cdot J}{\frac{U}{\ell}}}\right) \]

       4. *-commutative 61.1%

          \[\leadsto U \cdot \left(1 + \frac{\color{blue}{J \cdot 2}}{\frac{U}{\ell}}\right) \]

       5. associate-/l* 62.0%

          \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{2}{\frac{U}{\ell}}}\right) \]

   14. Simplified 62.0%

       \[\leadsto U \cdot \left(1 + \color{blue}{J \cdot \frac{2}{\frac{U}{\ell}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-239}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{2}{\frac{U}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.0% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2.2 \cdot 10^{-236}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2.2e-236)
   (* J (+ (* 2.0 l) (/ U J)))
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2.2e-236) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (k <= 2.2d-236) then
        tmp = j * ((2.0d0 * l) + (u / j))
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2.2e-236) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 2.2e-236:
		tmp = J * ((2.0 * l) + (U / J))
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 2.2e-236)
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 2.2e-236)
		tmp = J * ((2.0 * l) + (U / J));
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 2.2e-236], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 2.19999999999999992e-236

   1. Initial program 83.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 64.8%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 64.8%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 64.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 67.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 71.4%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 71.4%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 55.5%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 59.3%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 59.3%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   12. Taylor expanded in J around inf 55.7%

       \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]

   if 2.19999999999999992e-236 < K

   1. Initial program 92.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 61.1%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 61.1%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. associate-*r* 61.1%

         \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   5. Simplified 61.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

   6. Taylor expanded in U around inf 66.5%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 70.1%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 70.1%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 58.3%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 62.0%

          \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   11. Simplified 62.0%

       \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 2.2 \cdot 10^{-236}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.0% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(2 \cdot \ell\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* J (* 2.0 l))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (2.0 * l));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (2.0d0 * l))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (2.0 * l));
}

Python

def code(J, l, K, U):
	return U + (J * (2.0 * l))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(2.0 * l)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (2.0 * l));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 63.3%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. associate-*r* 63.3%

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   2. associate-*r* 63.3%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

5. Simplified 63.3%

   \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

6. Taylor expanded in K around 0 54.2%

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

   1. *-commutative 54.2%

      \[\leadsto U + \color{blue}{\left(J \cdot \ell\right) \cdot 2} \]

   2. associate-*r* 54.2%

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

8. Simplified 54.2%

   \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot 2\right)} \]

9. Final simplification 54.2%

   \[\leadsto U + J \cdot \left(2 \cdot \ell\right) \]
10. Add Preprocessing

Alternative 13: 37.1% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 86.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 63.3%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. associate-*r* 63.3%

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   2. associate-*r* 63.3%

      \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

5. Simplified 63.3%

   \[\leadsto \color{blue}{\left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + U \]

6. Taylor expanded in J around 0 36.9%

   \[\leadsto \color{blue}{U} \]

7. Final simplification 36.9%

   \[\leadsto U \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))